False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Disconnected elements need not lie in identity-component tori

Statement

Every element of a disconnected compact Lie group lies in a maximal torus of its identity component.

Facts & Assumptions

Given: The group $G : = Z / 2 = {0, 1}$ with the discrete topology, and the definition of a torus.

[L1]

A torus is a compact connected abelian Lie group, and a maximal torus of a compact Lie group is a torus subgroup maximal under inclusion; a connected subgroup of a discrete group is a singleton (Tori and maximal tori).

[L2]

$Z / 2$ is a compact Lie group of dimension zero; its identity component is the singleton ${0}$ , and the element $1$ is different from $0$ (Lie group, Tori and maximal tori).

Refutation

technique · direct

1.1

The group $Z / 2$ is discrete, so it is a compact zero-dimensional Lie group; its identity component is the connected component of $0$ , which is the singleton ${0}$ , and the element $1 \neq 0$ is not in it.

1.2

Every connected subgroup of the discrete group $Z / 2$ is a singleton by [L1], so the only torus contained in the identity component is the trivial torus ${0}$ ; it is the unique maximal torus of the identity component.

2.1

The nonidentity element $1$ does not lie in ${0}$ , so it lies in no torus of the identity component; hence the asserted statement fails already for the compact Lie group $Z / 2$ , and connectedness of the ambient group is a necessary hypothesis for torus-containment theorems.

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)